Rectification in Nonequilibrium Parity Violating Metamaterials

Understanding mechanisms for rectifying stochastic fluctuations has been a long-standing problem in nonequilibrium statistical mechanics. Here, we explore an opportunity provided by nonequilibrium parity-violating metamaterials to uncover new mechanisms for rectification of energy and motion. Using a parity-violating gyroscopic metamaterial that is allowed to interact with a bath of active particles as a model system, we develop an analytic diagrammatic theory that compactly elucidates how the rectification results from an interplay between gyroscopic forces, nonequilibrium activity, and network structure. Our active metamaterial model can generate energy flows through an object in the absence of any temperature gradient. The nonreciprocal microscopic fluctuations responsible for generating the energy flows can further be used to power locomotion in, or exert forces on, a viscous fluid. Taken together, our analytical and numerical results elucidate how the geometry and interparticle interactions of the parity-violating material can couple with the nonequilibrium fluctuations of an active bath and enable rectification of energy and motion.

Popular Summary

Identifying mechanisms that rectify stochastic fluctuations is a long-standing and important problem in nonequilibrium statistical mechanics. While studies of biological motor proteins and other related advances have shown how stochastic fluctuations can be rectified to power-directed motion, directed energy transport remains less well explored. Here, using a model metamaterial, we explore strategies to achieve directed energy transport, even in the absence of temperature biases.

Our platform is a spring-mass-type network model where the particles are further subject to magnetic forces and are allowed to interact with a bath of active energy-consuming particles. Our analytical and numerical results elucidate how the geometry of the network, the magnetic field, and the interactions with the active bath can couple to support net energy transport between sites even in the absence of any temperature gradients.

Together, our results can provide new strategies for rectification of energy and motion at the microscale, as well as a theoretical framework to understand similar rectification processes.

Figure 1

The model and the energy flux in example networks. (a) Schematic of the model, a spring-mass network subject to a Lorentz-like force and immersed in an active bath. (b)–(d) Averaged energy flux from numerical calculations for example networks. The flux direction and pattern can be controlled by the network geometry. In these figures, gray lines and dots represent the mechanical equilibrium structure of the network, and the arrows represent the flux between particles, whose length is proportional to the flux magnitude. An arrow is colored blue if it is counterclockwise (CCW), red if clockwise (CW), and gray for fluxes not on the boundary. The numerical calculations were performed with all parameters set to 1.

Figure 2

Necessary ingredients for generating nonzero energy fluxes. Both the Lorentz-like force and colored noise are needed to generate nonzero fluxes. The role of the Lorentz-like force is to provide chiral Fourier modes ($J^{FT}(\omega )\ne 0$). If $B=0$, $J^{FT}(\omega )$ is zero everywhere. The role of colored noise is to provide weighted excitation $h(\omega )\ne \text{const}$, which makes a nonvanishing averaged flux $⟨J⟩$ possible. The numerical calculations were performed with $m$, $k_g$, $k$, $\gamma$, $T_a=1$.

Figure 3

Illustrations of our diagrammatic technique. (a) Flux from site 1 to site 2 can be calculated by summing over diagrams. Each diagram is a closed path, which pictorially represents one term in the small-$k$ expansion. Paths are depicted using green arrows. The magnitude of the path of length $n$ is on the order of $k^n$. (b) A useful property of the diagram is that if there is no loop on a node, then the diagram can be simplified by removing all other branches on the node. (c) Schematic of a polygon path and its outer angles ${\theta }_1,{\theta }_2,\dots ,{\theta }_n$. The value of this diagram is written in Eq. (12). (d) The lowest-order flux from 1 to 2 in complex networks is the sum over the shortest cycles that contain $1\to 2$, which in this schematic are two pentagon diagrams ($k^5$). If, for instance, one of the pentagons is replaced by a triangle, then the triangle diagram ($k^3$) dominates the flux from 1 to 2.

Figure 4

Driving energy through a passive chain. (a) Conventional energy transport in a passive material (boxed in gray) with temperature differences at two ends. Similarly, the active gyroscopic network can also drive energy flows through a passive segment. (b) Instantaneous energy flux $J$ through bonds in the passive segment from a simulation. Fluxes through different bonds are colored differently. In general, $J$ is stochastic; however, during the period when $J$ is large, $J$ exhibits successive peaks in accordance with the direction of the flux. In the simulation setup, parameters for the active particles are as follows: $m$, $k_g$, $\gamma =0.1$, $k=10$, $\widehat{B}$, $T_a$, $\tau =1$. Passive particles are constrained to 1D, and their $\gamma$, $T_a$, $k_g$ are set to 0.

Figure 5

Utilization of nonequilibrium gyroscopic dynamics to power swimming and pumping in low $Re$ media. (a) The energy flux in the active gyroscopic network is accompanied by nonreciprocal motions of the particles (i,ii). The nonreciprocal motion is a schematic for illustration purposes, and the real data are much noisier. Using these nonreciprocal motions as input, a three-bead linear object can be made to swim through a low $Re$ medium (iii). Finally, by immersing the passive segment into a low $Re$ fluid, the nonreciprocal motions can be used to pump the fluid. In this manner, the energy fluxes can be rectified for locomotion and force generation (iv). (b) Swim speed $V_s$ is proportional to the energy flux $⟨J⟩$ in the active network. The proportionality constant is $-k_0/k$, which is independent of the network geometry. The series of dots for each $-k_0/k$ are obtained by varying the angles labeled by red disk sectors in pentagon networks or “$\mathrm{square}+\mathrm{tail}$” networks. The parameters chosen for the numerical calculations are $m$, $k_g=0.1$, $\tau =1$, $k=5$ for $k_0/k=0.04$ and $k=10$ for $k_0/k=0.02$. For the active part, $\gamma =0.1$, $\widehat{B}$, $T_a=1$. For the passive segment, $\gamma$, $\widehat{B}$, $T_a=0$.

Figure 6

Comparison between a boundary-localized eigenmode of the undamped isolated network and the Fourier modes of the damped network at the same frequency. First-order dynamics (by setting $m=0$) are used. Numerical calculations are performed with all other parameters set to 1. (a) Eigenmode of undamped gyroscopic system. For the frequency chosen, the eigenmode is localized on the boundary. Blue disks represent the orbit of particles. (b) The Fourier mode of the damped variant of our model at small $\gamma$ ($\gamma =0.001$) resembles the eigenmode. (c) The Fourier mode at larger $\gamma$ ($\gamma =1$) is no longer close to the eigenmode.

Figure 7

Using the eigenmode decomposition, we explain how the flux in the honeycomb network is CCW, even though its edge modes contribute to CW fluxes. (a) Network used for calculation, which consists of one row of hexagons (51 unit cells) and has a periodic boundary in the $x$ direction. The parameters are $k_{g,\tau }=1$, $k=10$, with others being 1. (b) Band structure of the network (marked with different colors). The yellow or green band contains CW flux localized on the top or bottom edge [an example mode is shown in panel (d) or (e)]. The blue band contains bulk modes with CCW flux [also see panel (f)]. (c) Weighted flux $J_{{\omega }_e}^{\mathrm{eig}}$ from 1 to 2 [marked in panel (a)] of the four bands. The total flux in the green band with CW edge modes and that in the blue band with CCW bulk modes are $-0.106$ and 0.115, respectively. As a result, the net flux is CCW.

Figure 8

Higher-order diagrams in a tailed-square network. (a) Direction of the flux can be controlled by the orientation of the side chain. (b) From the diagrammatic approach, the flux of the lowest-order diagram (square) vanishes, and the first nonvanishing diagram is affected by the side chain.

Figure 9

Decay of fluxes away from the boundary of honeycomb networks, and explanation using the diagrammatic technique. (a) Schematic of a honeycomb network that is periodic in the $x$ direction. Layers from the boundary are indexed as $n_l$. (b) Semilog plot of flux $⟨J⟩$ at layer $n_l$. The flux starting from layer $n_l=1$ shows exponential decay, with decay length $d_l$. Parameters used for numerical calculations are $\theta =\pi /6$, $k/k_0=0.01$, $\alpha =\pi /4$. (c) Decay length $d_l$, which changes with the network angle $\theta$ nonmonotonically, and the curve, which has a cusp at $\theta =\alpha =\pi /4$. At small $k/k_0$, perturbation theory results agree with numerical calculations. (d) The first nonvanishing diagram pair for $n_l=1$ with length 7. The two diagrams do not cancel because the loop in the bulk and that at the boundary have different values.

Figure 10

An example of the $⟨J_{31}^s⟩$ diagram (red) and the $⟨J_{12}^s⟩$ diagram (blue). Passive particles are boxed in gray, and the active ones are boxed in red.